Distance, similarity and entropy measures of n-dimensional fuzzy sets

An $n$-dimensional fuzzy set generalizes other fuzzy structures and effectively addresses real-world problems by providing greater flexibility in assigning membership values through the selection of an arbitrarily large $n$. Information measures are essential tools that yield significant judgments about data expressed in fuzzy form. This paper presents the concepts of $n$-dimensional distance measures, similarity measures, and entropy measures, accompanied by significant examples for each, and demonstrates the interrelationships among these measures. Certain specialized measures, including $\sigma$-measure, proximity measure, and linear measure, are examined, and significant results pertaining to them are derived. A succinct approximation of $n$-dimensional fuzzy sets is shown through the distance measure and the notion of orderless n-dimensional fuzzy sets, which proves advantageous in addressing practical issues. Ultimately, two decision-making dilemmas are resolved utilizing the concepts presented.